Cores in temporal networks A longitudinal approach

Cores in temporal networksA longitudinal approach

Vladimir Batagelj

Institute of Mathematics, Physics and Mechanics, Jadranska 19,1000 Ljubljana, Slovenia,

IAM, University of Primorska, 6000 Koper, Slovenia,ANR Lab High School of Economics, Moscow, Russia

Monika Cerinšek

Abelium d.o.o. R&D, Kajuhova 90, 1000 Ljubljana, Slovenia

November 7, 2021

AbstractAmong many formalizations of the notion of dense subnetworks of a

given network (clique, s-plexes, LS sets, lambda sets, cores, etc.) onlyfor cores exists an efficient algorithm that can compute the result in anacceptable time also for large networks.

In a graph G = (V, E) a subgraph H = (C, E(C)) induced by thesubset of nodes C is a k-core or a core of order k iff for each node v in Cits internal degree degH(v) is greater or equal to k, and H is a maximalsubgraph with this property.

Various generalizations of ordinary cores according to types of net-works and measures of node importance were proposed. In our contribu-tion we extend the notion of cores to temporal networks and present thecorresponding algorithms. For illustration we present results of determin-ing cores in selected artificial and real-life networks.Keywords: large network, temporal network, core, efficient algorithm,temporal quantityMathematics Subject Classification: 91D30; 05C22; 05C82; 05C85;62R07;

1 IntroductionThere are many formalizations of the notion of dense subnetwork(s) of a givennetwork (cliques, s-plexes, LS sets, lambda sets, cores, etc. [22]). In most casesthe corresponding algorithms are too expensive (require too much time) to beapplied to large networks. A very efficient algorithms exists for determiningcores. Besides this, some other interesting types of subnetworks are contained

1

in cores – cores can be used for efficient localization of such subnetworks [6, ].This makes cores very attractive and useful.

The notion of k-core was introduced in 1983 by Seidman[20]. A linear innumber of links algorithm was presented by Batagelj and Zaveršnik in 2003[9, 11, 6] and extended to other node property functions beside degrees [10, 6].In [1, 13] the (p, q)-cores were proposed as an extension of the idea to two-modenetworks. In this paper we present a further extension to temporal networksbased on the notion of temporal quantities [5].

Different ideas and applications of cores were exaustively reviewed in a recentpaper [18]. There are some recent works on temporal cores that can be classifiedin two groups: (1) core maintenance in networks changing by a stream of events[17, 23, 24, 4, 3, 21, 24, 2]; (2) the notion of a span-core [16].

Our approach differs because we use for each node a temporal quantitydescribing changes of its core number through time. In the paper we extend thenotion of a p-core to temporal networks and present the algorithms to determinethe temporal cores based on degrees and weighted degrees.

In Section 2 we present definitions of basic notions used in our method(temporal network, temporal core). In Section 3 we present the algorithm forour method. In Section 4 some results obtained in usage of our method ofartificial and real-life data are presented.

2 Definitions2.1 CoresA network N = (V,L,P,W); n = |V|, m = |L| is based on four sets: the set ofnodes V, the set of links L, the set of node properties P, and the set of weightsW. Each link is linking two nodes – its end-nodes. A link is either directed –an arc, or undirected – an edge. A pair of sets (V.L) forms a graph.

The notion of a k-core was introduced by Seidman in 1983 [20]: Let G =(V,L) be a graph with set of nodes V and set of links L. The degree of nodev ∈ V is denoted by deg(v). For a given integer k, a subgraph Hk = (Ck,L|Ck)induced by the subset of nodes Ck ⊆ V is called a k-core or a core of order k iffdegHk

(v) ≥ k, for all v ∈ Ck, and Hk is the maximum such subgraph.The core of maximum order is called the main core. The core number ,

core(v), of node v is the highest order of a core that contains this node.Let N(v) denote the set of neighbors of a node v in a network N , and

N(v, C) = N(v) ∩ C is the set of neighbors of a node v within the subset ofnodes C ⊆ V.

A very efficient algorithm exists for cores. All nodes have degree at least0. They belong to the core H0 of order 0. Removing all nodes of degree 0we obtain the set C1 in which each node has degree at least 1 – the core H1of order 1. Removing all nodes of (current) degree 1 we obtain the set C2 inwhich each node has degree at least 2 – the core H2 of order 2. . . . This leadsto an algorithm for determining each node’s core number core[v] presented inAlgorithm 1. An efficient, O(|L|), implementation of this algorithm is given in[6].

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Algorithm 1: Core decomposition algorithm.

1 CoreDecomposition (N ) :2 deg = {u : deg(u) for u ∈ V}3 C = V ; k = 1 ; core = { }4 while ¬empty(C) :5 while ∃u ∈ C : deg[u] < k :6 for v ∈ N(u, C) :7 deg[v] = deg[v]− 18 C = C \ {u}9 core[u] = k − 1

10 k = k + 111 return core

2.2 Generalized coresAssume that in a network N = (V,L,P,W) a node property function p(v, C) isdefined, p : V ×2V → R, where C ⊆ V is a cluster – a subset of nodes, and v ∈ Cis a node.

Some examples of such node properties were proposed in [6]:

1. p1(v, C) = degC(v): node degree within the induced subgraph N (v, C) =(C,L(C));

2. p2(v, C) = indegC(v)+outdegC(v): if all links are directed it holds p2 = p1;

3. p3(v, C) = wdeg(v, C) =∑

u∈N(v,C) w(v, u) for w : L → R+0 : weighted

degree, the sum of weights of incident links within N (v, C);

4. p4(v, C) = maxu∈N(v,C) w(v, u) for w : L → R: the maximum weight ofincident links within N (v, C);

5. p5(v, C) = degC(v)deg(v) if deg(v) > 0 else f5(v, C) = 0: the fraction of neighbors

within N (v, C);

6. p6(v, C) = wdeg(v,C)wdeg(v,V) for w : L → R+

0 : the fraction of the weighted degreeof incident links within N (v, C).

The node property functions are used in the definition of generalized cores[6].

The subgraph H = (C,L(C)) induced by the set C ⊆ V is a p-core at levelt ∈ R iff ∀v ∈ C : t ≤ p(v, C) and C is the maximum such set.

We say that the node property function p(v, C):

• is local iff: p(v, C) = p(v, N(v, C)) ∀v ∈ V.

• is monotonic iff: C1 ⊂ C2 ⇒ ∀v ∈ V : p(v, C1) ≤ p(v, C2)..

For a local and monotone property function p the corresponding p-core numberscan be efficiently determined using a generalized version of Algorithm 1 [6].

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2.3 Temporal network and temporal coreWe get a temporal network NT = (V,L, T ,P,W) by attaching the time T toan ordinary network, where T = [Tmin, Tmax) is a linearly ordered set of timepoints t ∈ T which are usually integers or reals. In a temporal network activity(presence / absence) of nodes and links may change through time. Let TV (v),TV ∈ P, be the activity set of time points for node v ∈ V and TL(`), TL ∈ W,the activity set of time points for link ` ∈ L. Consistency condition must befulfilled for activity sets: If a link `(u, v) is active at the time point t then itsend-nodes u and v should be active at the time point t :

TL(`(u, v)) ⊆ TV (u) ∩ TV (v).

Beside this also the values of node properties and link weights may change.These changes can be described in different ways. In this paper we will use theapproach based on temporal quantities (TQs) [5, 8].

A temporal quantity a with the activity set Ta ⊆ T describes the changes ofits value through time:

a ={

a(t) t ∈ Ta;t ∈ T \Ta.

We assume that the values of temporal quantities belong to a set A which is asemiring (A, +, ·, 0, 1) for binary operations + : A × A → A and · : A × A →A. In this paper we will limit our attention to the combinatorial semiring(R+

0 , +, ·, 0, 1) where + is the addition and · is the multiplication of numbers.We can extend both operations to the set A = A ∪ { } by requiring that forall a ∈ A it holds

a + = + a = a and a · = · a =

The structure (A , +.·, , 1) is also a semiring. Let us denote A (T ) the set ofall temporal quantities over A in time T . To extend the operations to networksand their matrices we first define the sum (parallel links) a + b as

(a + b)(t) = a(t) + b(t) and Ta+b = Ta ∪ Tb.

and the product (sequential links) a · b as

(a · b)(t) = a(t) · b(t) and Ta·b = Ta ∩ Tb.

Nodes and links in temporal networks have temporal quantities assignedas values of some properties/weights. For cumputational reasons we limit ourfurther discussion to temporal quantities or TQs that are represented with alist of triples (s, f, v) and each triple determine the time interval [s, f) in whichnode or link is active or present, and its value v is constant on this time interval.

An example of temporal network is shown in Figure 1. It is defined on adiscrete time set T = [1, 9) = {1, 2, 3, 4, 5, 6, 7, 8}. All nodes in this networkare present through whole time interval. Some links are directed – direction isindicated with an arrow. All links have value 1 when they are present. Somelinks are not present through all the time. For example, link from node 2 tonode 4 is present in time points [3, 9) = {3, 4, 5, 6, 7, 8}.

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A time slice N (t) = (V(t),L(t),P(t),W(t)) of the temporal network NT

in a time point t consists of nodes and links active in a time point t and thecorresponding values of node properties and link weights in a time point t. Thenotion of a time slice can be extended to a time interval I ⊂ T by

N (I) =⋃t∈IN (t)

The p-core number p-core(v, t) of node v ∈ V in time point t ∈ TV (v) isequal to the p-core number of node v in the time slice N (t); and is equal tootherwise. It determines a TQ p-core(v).

3 Algorithms3.1 Temporal degree coresThe algorithm for temporal core decomposition is based on algorithm for k-coredecomposition that is presented in Alg. 1.

In the description of the algorithm we will use Python-like comprehensionexpressions for lists

L = [E(s) for s ∈ S if P (s)]

and for dictionaries

D = {s : E(s) for s ∈ S if P (s)}

S is a sequence (list, set, dictionary), s is an item from the sequence; E(s) is anexpression transforming the item s into a value.

• TQsum(a, b) – returns the sum a + b of TQs a and b;

• TQminus(a) – returns the TQ −a;

• TQsetConst(a, k) – sets the value component in each triple of the TQ ato k;

• TQmin(a) – returns minimum of values in triples of the TQ a; returns ∞

• TQcutGE(a, k) – removes from the TQ a all triples tq with tq.v < k;

• TQcutGT (a, k) – removes from the TQ a all triples tq with tq.v ≤ k;

• TQcutLE(a, k) – removes from the TQ a all triples tq with tq.v > k;

• TQcutEQ(a, k) – removes from the TQ a all triples tq with tq.v 6= k;

• TQlower(a, k) – returns the TQ a in which values smaller than k are setto k;

• TQextract(a, b) – extracts a from b – restricts the TQ b to the activityset Ta of the TQ a;

• TQcomplement(a, tmin, tmax) – returns the TQ with triples determinedby the activity set [tmin, tmax)\Ta with the value component equal to 1;

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• TQdeg(v) – returns the temporal degree of the node v ∈ V;

• TQwdeg(v) – returns the temporal weighted degree of the node v ∈ V

wdeg(v) =∑

`∈star(v)

w(`)

D is a dictionary of (still active) node temporal degrees. D[u] is a temporaldegree of the node u ∈ V. Core is a dictionary of currently known temporalcore numbers. Core[u] is a TQ with temporal core numbers of the node u ∈ V.

tq is a triple from a TQ. It has components: tq.s – start, tq.f – finish, andtq.v – value.

Algorithm 2: Degree cores in temporal network.

1 TemporalDegreeCores (N ) :2 D = {u : TQdeg(u) for u ∈ V}3 Core = {u : TQcutLE(D[u], 0) for u ∈ V}4 D = {u : TQcutGT (D[u], 0) for u ∈ V}5 D = {u : D[u] for u ∈ V i f ¬empty(D[u])}6 Dmin = {u : TQmin(D[u]) for u ∈ D}7 while ¬empty(D) :8 u = argmin(Dmin) ; dmin = Dmin[u]9 core = [tq for tq ∈ D[u] i f tq.v = dmin]

10 Core[u] = TQsum(Core[u], core)11 change = TQsetConst(core,−1)12 D[u] = TQcutGE(TQsum(D[u], change), dmin)13 i f empty(D[u]) : delete(D, u) ; delete(Dmin, u)14 else : Dmin[u] = TQmin(D[u])15 for ` in star(u) :16 v = twin(u, `)17 i f v /∈ D : continue18 chLink = TQextract(w(`), change)19 i f empty(chLink) : continue20 diff = TQcutGE(TQsum(D[v], chLink), 0)21 i f empty(diff) : delete(D, v) ; delete(Dmin, v)22 else :23 D[v] = TQlower(diff, dmin)24 Dmin[v] = TQmin(D[v])25 return Core

DELETE ???We expanded Alg. 1 onto temporal quantities and the new algorithm is shown

in Alg. 3. Input for this algorithm is temporal network. Before main loop wecopy node temporal quantities into new set D, define set of core numbers fornode in shape of temporal quantities, removed temporal quantities from D thathave a degree value equal to 0, and defined the dictionary Dmin which consistsof keys equal to nodes and values equal to minimum degree of a node. This isshown in lines 2-5 in Alg. 3.

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The main loop (lines 6-29 in Alg. 3) is repeated until the set D is not empty.At first we find a pair (dmin, u) from Dmin where dmin is minimum of all valuesin Dmin. As core we define set of temporal quantities from D[u] with degreeequal to dmin. We add these temporal quantities to the set of core numbers(line 11 in Alg. 3) as their degree value represent core numbers for node u ingiven time intervals. We rewrite core into change with degree values equal to−1. This we need to save only those temporal quantities in D[u] that have adegree larger than dmin where we reduce their degrees for 1 in those that arealso in change.

Now we can check all neighbours of node u – we loop through all links ofu – lines 14-25 in Alg. 3. If the other end-node of a link is not in D, we skipit and check another link. Otherwise we define changeLink as intersection oftemporal quantities of a link and change : time intervals are intersections of timeintervals, value is equal to −1. Link changeLink is actually active the same timeas u with degree value equal to dmin. If this link is never active, we skip it andcheck another link neighbouring u. Now we define diff as temporal quantitiesfrom D[v] with degree value more than 0, where we reduce their degrees for 1in those that are also in changeLink. Now we rewrite D[v] as diff with degreevalues equal to max(deg, dmin). If D[v] is empty, we delete node v from D andDmin. Otherwise we set a degree value for Dmin[v] to be equal to minimumdegree value in D[v].

After checking all neighbouring links of node u we check if there are anynon-empty temporal quantities in D[u] – lines 26-29 in Alg. 3. If there are none,we delete node u from D and Dmin. Otherwise we set a degree value for u inDmin to be minimum degree value in temporal quantities of D[u].

*** Correctness check an instance t

3.1.1 Example

The network is shown in Fig. ?? Figure 5 from [5]. It consists of 15 nodes thatare always present. Most of the links are directed, some are undirected. Mostof the links are present all the time T = [1, 9) = {1, 2, 3, 4, 5, 6, 7, 8}, Tmin = 1and Tmax = 9 – for them there are no temporal quantities written in figure.Links (2, 4), (5, 7), (11, 7), (13, 14), (13, 15), (14, 15) are present in defines timeintervals. All links have value equal to 1 when they are active. The degrees ofnodes are listed in the left side of Tab. 1. For example, node 13 has a degreeequal to 0 in the time interval [1, 2) (time point 1) because both neighbouringlinks are not present in this time interval. In time interval [2, 8) are both linkspresent, so the node has a degree equal to 2. After that its degree is again equalto 0, because no incident link is active.

In the algorithm for temporal core decomposition we need to define sets D,CoreHierarchy, and Dmin. For this examples these are:

• D = {1 : [(1, 9, 1)], 2 : [(1, 3, 2), (3, 9, 3)], ..., 15 : [(1, 2, 0), (2, 8, 2), (8, 9, 0)]}

• CoreHierarchy = {12 : [(1, 9, 0)], 13 : [(1, 2, 0), (8, 9, 0)], 14 : [(1, 2, 0),(8, 9, 0)], 15 : [(1, 2, 0), (8, 9, 0)]}

• Remove temporal quantities with value 0 from D: D = {1 : [(1, 9, 1)], 2 :[(1, 3, 2), (3, 9, 3)], ..., 15 : [(2, 8, 2)]}

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Figure 1: First example network. All unlabeled links have a value of [(1, 9, 1)].

• Dmin = {1 : 1, 2 : 2, 3 : 1, 4 : 2, 5 : 2, 6 : 2, 7 : 3, 8 : 4, 9 : 4, 10 : 4, 11 :3, 12 : 0, 13 : 0, 14 : 0, 15 : 0}

We determine dmin from Dmin to be 0, which is first core number wecalculate cores with. The result of the algorithm is shown on the right in Tab. 1.Core number for node 1 is equal 1 all the time. Core number for node 15 isequal to 0 in time intervals [1, 2) and [8, 9), and is equal to 2 in time interval[2, 8).

3.2 Temporal weighted degree corespS-cores are very similar to ordinary cores. For ordinary cores the node propertyfunction is p1 = deg, and for pS-cores the node property function is p3 = wdeg.Note that in the case w(`) = 1 for all ` ∈ L p3 = wdeg = deg = p1. Thereforethe pS-cores algorithm can be used also for computing ordinary cores.

We can base our algorithm for pS-cores on Alg.2 with some adaptations forchanges of the values of property function.

In application of Algorithm 3 to some bibliographic networks based on frac-tional approach [] it turned out that it can have problems with rounding errors(*** example). To account for them we introduced the precision parameter eps(for example, eps = 10−5) and replace lines 10 and 19 with

10 core = TQextract(cCore, [tq for tq ∈ D[u] i f tq.v ≤ dmin + eps])

and

19 diff = TQcutGE(TQsum(D[v], chLink),−eps)

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Table 1: Degrees and degree core numbers for first artificial network.

Node Degree Degree core number1 (1, 9, 1) (1, 9, 1)2 (1, 3, 2), (3, 9, 3) (1, 9, 1)3 (1, 9, 1) (1, 9, 1)4 (1, 3, 2), (3, 9, 3) (1, 9, 2)5 (1, 5, 3), (5, 9, 2) (1, 9, 2)6 (1, 9, 2) (1, 9, 2)7 (1, 5, 4), (5, 7, 3), (7, 9, 4) (1, 7, 3), (7, 9, 4)8 (1, 9, 4) (1, 7, 3), (7, 9, 4)9 (1, 9, 4) (1, 7, 3), (7, 9, 4)

10 (1, 9, 4) (1, 7, 3), (7, 9, 4)11 (1, 7, 3), (7, 9, 4) (1, 7, 3), (7, 9, 4)12 (1, 9, 0) (1, 9, 0)13 (1, 2, 0), (2, 8, 2), (8, 9, 0) (1, 2, 0), (2, 8, 2), (8, 9, 0)14 (1, 2, 0), (2, 8, 2), (8, 9, 0) (1, 2, 0), (2, 8, 2), (8, 9, 0)15 (1, 2, 0), (2, 8, 2), (8, 9, 0) (1, 2, 0), (2, 8, 2), (8, 9, 0)

Algorithm 3: Weighted degree cores in temporal network.

1 TemporalWeightedDegreeCores (N ) :2 D = {u : TQwdeg(u) for u ∈ V}3 Core = {u : TQcutLE(D[u], 0) for u ∈ V}4 D = {u : TQcutGT (D[u], 0) for u ∈ V}5 D = {u : D[u] for u ∈ V i f ¬empty(D[u])}6 Dmin = {u : TQmin(D[u]) for u ∈ D}7 while ¬empty(D) :8 u = argmin(Dmin) ; dmin = Dmin[u]9 cCore = TQcomplement(Core[u], Tmin, Tmax)

10 core = TQextract(cCore, TQcutLE(D[u], dmin))11 i f ¬empty(core) :12 Core[u] = TQsum(Core[u], core)13 D[u] = TQcutGE(TQsum(D[u], TQminus(core)), dmin)14 for ` in star(u) :15 v = twin(u, `)16 i f v /∈ D : continue17 chLink = TQminus(TQextract(core, w(`)))18 i f empty(chLink) : continue19 diff = TQcutGE(TQsum(D[v], chLink), 0)20 D[v] = TQlower(diff, dmin)21 i f empty(D[v]) : delete(D, v) ; delete(Dmin, v)22 else : Dmin[v] = TQmin(D[v])23 i f empty(D[u]) : delete(D, u) ; delete(Dmin, u)24 else : Dmin[u] = TQmin(D[u])25 return Core

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3.2.1 Example

We defined another artificial temporal network (Fig. 2) with the same underlyinggraph structure as in previous network (Fig. ??). The presence and values oflinks are different which is shown in Fig. 2 and in values of nodes in Tab. 2. Weselected sum of neighbouring link values as node values for this example so wecan calculate pS-core numbers for nodes. The result of the algorithm is shownon the right side of Tab. 2.

Figure 2: Visualization of second artificial example of a temporal network. Asingle value v on an arc is an abreviation for [(1, 9, v)]

4 ResultsWe tested our method on artificial examples and examples from real-life withdifferent context. Firstly we will look at artificial example to better understandwhat temporal network is and how do we determine temporal cores.

4.1 Reuters terror news networkThe Reuters terror news network was obtained using the CRA (Centering Reso-nance Analysis) by Steve Corman and Kevin Dooley at Arizona State University[14]. This temporal network is available at

http://vlado.fmf.uni-lj.si/pub/networks/data/CRA/terror.htm.The source data are all stories released during 66 consecutive days by the

news agency Reuters concerning the September 11 attack on the U.S. This timeinterval starts at 9:00 AM EST 11th of September 2001.

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Table 2: Weighted degrees and weighted degree core numbers for second artificialnetwork.

Node Weighted degree Weighted degree core number1 (1, 5, 3), (5, 9, 5) (1, 5, 3), (5, 9, 5)2 (1, 3, 7), (3, 9, 10) (1, 5, 4), (5, 9, 5)3 (1, 5, 4), (5, 9, 2) (1, 5, 4), (5, 9, 2)4 (1, 3, 4), (3, 9, 7) (1, 5, 4), (5, 9, 5)5 (1, 5, 10), (5, 9, 7) (1, 9, 5)6 (1, 9, 7) (1, 9, 5)7 (1, 5, 13), (5, 7, 10), (7, 9, 14) (1, 9, 10)8 (1, 5, 13), (5, 9, 10) (1, 9, 10)9 (1, 5, 19), (5, 9, 16) (1, 9, 10)

10 (1, 9, 11) (1, 9, 10)11 (1, 7, 11), (7, 9, 15) (1, 9, 10)12 (1, 9, 0) (1, 9, 0)13 (1, 2, 0), (2, 5, 6), (5, 8, 9), (8, 9, 0) (1, 2, 0), (2, 5, 5), (5, 8, 7), (8, 9, 0)14 (1, 2, 0), (2, 8, 7), (8, 9, 0) (1, 2, 0), (2, 5, 5), (5, 8, 7), (8, 9, 0)15 (1, 2, 0), (2, 5, 5), (5, 8, 8), (8, 9, 0) (1, 2, 0), (2, 5, 5), (5, 8, 7), (8, 9, 0)

Nodes of the network are important words / terms. Two terms are linked ifthey co-appear in the same text unit (sentence). The link value is equal to thefrequency of such appearances per day – the network is temporal with a daygranularity. There are n = 13332 nodes and m = 243447 edges in the network.50859 of edges have value larger than 1. There are no loops.

We determined the degree cores in the terror news network. The computa-tion took 1 minute and half on a standard PC. The maximum core number is17. How to present the results? There are 13332 TQs to be inspected.

One approach is to restrict our attention to cores with large core numbers –at least k ∈ N. We produced cuts Ck for k = 6, 10, 12, 15. The sizes (number ofnodes) of the obtained cuts are as follows: |C6| = 2282, |C10| = 494, |C12| = 243,|C15| = 34.

In Figure 3 the TQs from the cut C12 are presented as a heatmap. Theordering of its rows (terms) is determined by a hierarchical clustering, completemethod. For computing the dissimilarity between rows a special measure

d(x, y) = 1|Tx ∪ Ty|

∑t∈Tx∪Ty

|x[t]− y[t]|1 + max(0, x[t], y[t])

was used. In computing d we replace in TQs the undefined value with thevalue −1. Only columns (days) with at least some defined value are presented(Sep 11, Sep 12, Sep 16, Sep 26, Oct 11).

*** comment on core nums 17 and 16The cut C6 has 2282 nodes. Too much to apply the same approach as for

C12. We can still produce a clustering tree (dendrogram, see Figure 4), exportit in PDF, and browse it in a viewer (Marquee Zoom in Acrobat Reader) tovisually detect interesting clusters (subtrees in the dendrogram). We extractthe selected clusters and visualize the corresponding TQs as a heatmap.

*** Comments strongest Sep 26, first six days,

11

sep−11

sep−12

sep−16

sep−26

oct−11

terror

attack

new_york

strike

timor

sunday

ship−to−shore

revenge

rehearse

landing

helicopter

east

expectation

casualty

bomb

report

civilian

thursday

stronghold

respond

kandahar

grow

intense

kabul

ultimatum

rule

protester

fresh

embassy

desert

coalition

afghanistan

anti−terror

wednesday

capital

taliban

united_states

pres_bush

fire

set

undergrind

store

stockpile

site

salt

oil

million

mexico

gulf

four

deep

crude

coast

cavern

500

barrel

two

support

spokesman

schedule

saudi

response

reporter

price

possible

place

morgan_stanlly_dw

measure

los_angeles

leader

lawmaker

issue

investigation

interview

include

flag

firefighter

early

district

death

crime

controller

chief

car

assault

ashcroft

a.m.

action

war

victim

unprecedented

tragedy

system

south

service

search

rubble

police

percent

passenger

number

news

national

member

mayor_giuliani

limit

life

information

hijacker

ground

florida

fbi

clear

business

boston

big

bank

authority

act

airline

wreckage

west

trade

top

terrorism

tell

team

target

suspect

survivor

state

source

side

send

section

school

say

return

rescuer

rescue

point

personnel

pennsylvania

operation

officer

night

newspaper

new

minute

mayor

market

man

local

late

investigator

intelligence

inc

group

giant

floor

faa

employee

deadly

damage

control

congress

company

collapse

capitol

brief

bin_laden

airliner

aviation

street

huge

history

explosion

evacuation

cause

devastate

emergency

way

traffic

the_worst

smoke

blood

scene

terrorist

television

military

massive

jet

hospital

home

headquarters

federal

crash

agency

airplane

world_trade_ctr

world

worker

work

white_house

washington

twin

tuesday

tower

time

thousand

security

plane

people

pentagon

part

official

office

nation

morning

manhattan

major

low

large

landmark

hour

hijack

government

flight

financial

day

country

commercial

city

center

call

buildngs

buildng

area

american

america

airport

air

aircraft

12

13

14

15

16

17

sep−11

sep−12

sep−16

sep−26

oct−11

terror

attack

new_york

strike

timor

sunday

ship−to−shore

revenge

rehearse

landing

helicopter

east

expectation

casualty

bomb

report

civilian

thursday

stronghold

respond

kandahar

grow

intense

kabul

ultimatum

rule

protester

fresh

embassy

desert

coalition

afghanistan

anti−terror

wednesday

capital

taliban

united_states

pres_bush

fire

set

undergrind

store

stockpile

site

salt

oil

million

mexico

gulf

four

deep

crude

coast

cavern

500

barrel

two

support

spokesman

schedule

saudi

response

reporter

price

possible

place

morgan_stanlly_dw

measure

los_angeles

leader

lawmaker

issue

investigation

interview

include

flag

firefighter

early

district

death

crime

controller

chief

car

assault

ashcroft

a.m.

action

war

victim

unprecedented

tragedy

system

south

service

search

rubble

police

percent

passenger

number

news

national

member

mayor_giuliani

limit

life

information

hijacker

ground

florida

fbi

clear

business

boston

big

bank

authority

act

airline

wreckage

west

trade

top

terrorism

tell

team

target

suspect

survivor

state

source

side

send

section

school

say

return

rescuer

rescue

point

personnel

pennsylvania

operation

officer

night

newspaper

new

minute

mayor

market

man

local

late

investigator

intelligence

inc

group

giant

floor

faa

employee

deadly

damage

control

congress

company

collapse

capitol

brief

bin_laden

airliner

aviation

street

huge

history

explosion

evacuation

cause

devastate

emergency

way

traffic

the_worst

smoke

blood

scene

terrorist

television

military

massive

jet

hospital

home

headquarters

federal

crash

agency

airplane

world_trade_ctr

world

worker

work

white_house

washington

twin

tuesday

tower

time

thousand

security

plane

people

pentagon

part

official

office

nation

morning

manhattan

major

low

large

landmark

hour

hijack

government

flight

financial

day

country

commercial

city

center

call

buildngs

buildng

area

american

america

airport

air

aircraft

12

13

14

15

16

17

sep−11

sep−12

sep−16

sep−26

oct−11

terror

attack

new_york

strike

timor

sunday

ship−to−shore

revenge

rehearse

landing

helicopter

east

expectation

casualty

bomb

report

civilian

thursday

stronghold

respond

kandahar

grow

intense

kabul

ultimatum

rule

protester

fresh

embassy

desert

coalition

afghanistan

anti−terror

wednesday

capital

taliban

united_states

pres_bush

fire

set

undergrind

store

stockpile

site

salt

oil

million

mexico

gulf

four

deep

crude

coast

cavern

500

barrel

two

support

spokesman

schedule

saudi

response

reporter

price

possible

place

morgan_stanlly_dw

measure

los_angeles

leader

lawmaker

issue

investigation

interview

include

flag

firefighter

early

district

death

crime

controller

chief

car

assault

ashcroft

a.m.

action

war

victim

unprecedented

tragedy

system

south

service

search

rubble

police

percent

passenger

number

news

national

member

mayor_giuliani

limit

life

information

hijacker

ground

florida

fbi

clear

business

boston

big

bank

authority

act

airline

wreckage

west

trade

top

terrorism

tell

team

target

suspect

survivor

state

source

side

send

section

school

say

return

rescuer

rescue

point

personnel

pennsylvania

operation

officer

night

newspaper

new

minute

mayor

market

man

local

late

investigator

intelligence

inc

group

giant

floor

faa

employee

deadly

damage

control

congress

company

collapse

capitol

brief

bin_laden

airliner

aviation

street

huge

history

explosion

evacuation

cause

devastate

emergency

way

traffic

the_worst

smoke

blood

scene

terrorist

television

military

massive

jet

hospital

home

headquarters

federal

crash

agency

airplane

world_trade_ctr

world

worker

work

white_house

washington

twin

tuesday

tower

time

thousand

security

plane

people

pentagon

part

official

office

nation

morning

manhattan

major

low

large

landmark

hour

hijack

government

flight

financial

day

country

commercial

city

center

call

buildngs

buildng

area

american

america

airport

air

aircraft

12

13

14

15

16

17

sep−11

sep−12

sep−16

sep−26

oct−11

terror

attack

new_york

strike

timor

sunday

ship−to−shore

revenge

rehearse

landing

helicopter

east

expectation

casualty

bomb

report

civilian

thursday

stronghold

respond

kandahar

grow

intense

kabul

ultimatum

rule

protester

fresh

embassy

desert

coalition

afghanistan

anti−terror

wednesday

capital

taliban

united_states

pres_bush

fire

set

undergrind

store

stockpile

site

salt

oil

million

mexico

gulf

four

deep

crude

coast

cavern

500

barrel

two

support

spokesman

schedule

saudi

response

reporter

price

possible

place

morgan_stanlly_dw

measure

los_angeles

leader

lawmaker

issue

investigation

interview

include

flag

firefighter

early

district

death

crime

controller

chief

car

assault

ashcroft

a.m.

action

war

victim

unprecedented

tragedy

system

south

service

search

rubble

police

percent

passenger

number

news

national

member

mayor_giuliani

limit

life

information

hijacker

ground

florida

fbi

clear

business

boston

big

bank

authority

act

airline

wreckage

west

trade

top

terrorism

tell

team

target

suspect

survivor

state

source

side

send

section

school

say

return

rescuer

rescue

point

personnel

pennsylvania

operation

officer

night

newspaper

new

minute

mayor

market

man

local

late

investigator

intelligence

inc

group

giant

floor

faa

employee

deadly

damage

control

congress

company

collapse

capitol

brief

bin_laden

airliner

aviation

street

huge

history

explosion

evacuation

cause

devastate

emergency

way

traffic

the_worst

smoke

blood

scene

terrorist

television

military

massive

jet

hospital

home

headquarters

federal

crash

agency

airplane

world_trade_ctr

world

worker

work

white_house

washington

twin

tuesday

tower

time

thousand

security

plane

people

pentagon

part

official

office

nation

morning

manhattan

major

low

large

landmark

hour

hijack

government

flight

financial

day

country

commercial

city

center

call

buildngs

buildng

area

american

america

airport

air

aircraft

12

13

14

15

16

17

Figure 3: Terror news temporal cores at level 12.

Figure 4: Terror news temporal cores at level 6 – dendrogram.

In Figure 5 we present the cluster that contains the main "actors": united_states,washington, taliban, afghanistan, government. The corresponding TQs are dis-played as a heatmap in Figure 6.

The second selected cluster is built around the term anthrax (biologicalwarfare). It is presented in Figure 7 and the corresponding TQs in Figure 8.

12

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Figure 5: Terror news temporal cores at level 6 – cluster 1.

*** Comment senator, date; entered laterThe described approach can be used on up to some thousands of nodes.

Another approach to large sets of nodes is to consider only the most activenodes. The activity of the TQ a can be measured by its aggregated value [5]

Σa =∑

i

(fi − si) · vi

The procedure is now simple – order the nodes in decreasing order with respectto their activity and extract the g ∈ N top one. Display them with a heatmap.For terror news network cores, g = 30, it is presented in Figure 9.

*** but time span of these cores is wider, not limited mostly to first 11 days.The max pS-core numbers through time interval is shown in Fig. ??. It drops

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eyewitnesscross

evacuatesoldiertough

commandflash

controlairplane

trydept

heartseries

floorreturn

endsubway

directincident

commentyasser_arafat

increasefly

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tripcasualtyrespond

bloodcommuter

vehicledowntown

canadaolsonblow

evacuationhorrorinjurynerve

pearl_barborhusband

tallbad

dullesburn

shockwindow

tedsolicitor

getnewark

woodpanic

carnageimageyoung

presidencyhorrific

lightsafe

momentarlingtonlocation

pittsburghsecret

traumastore

georgewife

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educationagreement

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navalmultimillionaire

mountmind

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maneuverinside

identifyhard

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capabilitysympathy

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expectationhard−linebiological

weaponface

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ridleyarabic−language

sheltonpuritanical

okazex−king

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189ali

documentzerou.n

councilurgegrow

jamespanel

judiciaryleahy

koizumidiplomat

importantopinion

australiancallan

30agricultural

zhuwam

religionrahmanpatrickmentor

iranian−backedhidehiko

gatherfollower

delraycomputercallahanbangzao

bangladesh130

anti−terroristschieffer

satojakartafourth

howardvp_cheney

two−dayachakzai

nematullahanchorpound

nomguerre

deatty_gen_ashcroftsaid

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egyptiananthrax−laced

fierceairporthome

siteagent

systemconference

newstop

enforcementlaw

theywakebase

suicidewednesday

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informationsuspect

yearorganization

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officerfinancial

airlinehijacker

misspennsylvania

nationalnumberpercent

responsereuter

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namesenior

statementmeet

presidentarabway

majorintelligence

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workinclude

operationgroundcapitolsaudi

concernlate

casecivilianserviceairliner

thousandcall

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mayormayor_giuliani

fearnight

mediaperson

itfamily

twoinvestigation

biglink

deadlyflight

passengertower

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televisionsmall

assaulthour

morningbuildngs

companyfirst

eventbusiness

personnelinvestigatorcoordinate

united_airlinesamerican_airlines

ofdeathscene

damagesymbolboston

totalminute

faasafety

manhattanside

commerciallife

pilotpowell

actsurvivor

debris110−story

followvicefirmtrain

aviationloss

telephonelong

blockschedule

phoneboeing

celldozen

defense_deptsearch

hundredrescuerubble

emergencydevastate

headchicago

crewfirefighterlandmark

marketexchange

stockcrash

domesticannounce

parkarabianunusual

threesergeant

peninsulapecker

outcomehigh−rise

hangexpatriate

e−tradecontamination

brokeragebattery

asbestosapartment

10−year−oldami

davidson

onlinerelative

incexecutive

skytreasurycurrent

doubtactive

dutyshift

versionsensenbrenner

safariniplot

millenniumliberty

explosivechallenge

cast1.4

biodefensequadrennial

three−monthbrentwood

involvemortalitymaternal

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frontline11

northern_allianceopposition

0.0 0.4 0.8

Terror cores 6/Com

plete

hclust (*, "complete")

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Figure 9: Terror news temporal cores at all levels – most active.

rapidly after six days which is also consistent with the idea of news getting oldwith passing time.

4.2 Authors from the field of social network analysis till2018

As an example of application of temporal weighted degree cores we will analyzethe cummulative fractional collaboration (co-authorship) network obtained fromthe collection of networks on the topic of social network analyzed in the papers

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Table 3: Authors with the largest accumulated fractional valueTABELA

[19]. The networks were constructed from the Web of Science data. For detailssee the paper [19].

We start with the two-mode network WA (works× authors) and the vector ywhich contains for each work (paper, book, report) its publication year. Thereare 70792 works and 93011 authors. We combine them into a cummulativetemporal network WAtc = [watc(w, a)], where watc(w, a) = [(y(w), Tmax, 1)]for (w, a) ∈ LWA [8]. The TQ [(y(w), Tmax, 1)], the weight of the link (w, a),essentialy says that this link from the work w to its author a is active from theyear of publication of the work w on.

We extend the standard normalization (fractional approach)

n(WA)(w, a) =

and Newman’s normalization

n′(WA)(w, a) =

to temporal networks and using the temporal networks multiplication we finallyget the temporal cumulative co-authorship network

CNt = n(WAtc)T · n′(WAtc)

This is an undirected network with loops removed.We compute the weighted degree cores of the network CNt. It took 20

minutes on a standard PC. We get temporal weighted degree core numbers for93011 authors. Again, the obvious result is to display the most active authors.Since the values in TQs in cumulative co-authorship network are increasing wecan use as a measure of activity the accumulated fractional value in the lastyear. For g = 63 we get the list of authors in Table 3.

Figure 10 presents the heatmap for these authors.*** CommentsIf we would like to get more precise information about the TQs of selected

nodes we can present them as "histograms". In Figure 11 we compare the TQsfor the weighted degree cores for Borgatti and Pattison.

In (very) large networks most of the nodes are of small degree. If we are onlyinterested in nodes with large core numbers, say at least k, we can speed-up thecomputation by considering the property that all nodes in the k-core have degreeat least k. Therefore we can remove from the node degree data all intervals forwhich the node degree is less than k. We also remove all nodes with empty TQresulting from this transformation. We apply the temporal cores procedure onthe reduced network and apply the same reduction on the obtained temporalcore.

5 ConclusionTODO

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REZVANIA_A

BATAGELJ_V

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FERLIGOJ_A

MRVAR_A

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VALENTE_T

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ORTEGA_J

ARENTZE_T

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BRODKA_P

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VASILAKO_A

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CROFT_D

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CHELMIS_C

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LEYDESDO_L

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SKVORETZ_J

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Figure 10: SNA literature largest Ps cores.

Analyze the complexity of the algorithmImprove the complexity of the algorithmExtend the algorithm to generalized temporal coresFind user friendly presentations of resultsCompare with the streaming core algorithmsTemporal Quantities - a Python 3 library for temporal network analysis:

http://vladowiki.fmf.uni-lj.si/doku.php?id=tq

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Figure 11: SNA literature – comparing Borgatti and Pattison.

AcknowledgmentsWe thank the two anonymous reviewers whose comments and suggestions helpedimprove and clarify this manuscript.

All computations were performed using the program for large network anal-ysis and visualization Pajek [?] and the statistical programming system R.

This work was supported in part by the Slovenian Research Agency (researchprogram P1-0294 and research projects J1-9187 and J5-2557), and preparedwithin the framework of the HSE University Basic Research Program.

The second author was partially sponsored by Slovenian Research Agency(ARRS) - projects Z7-7614 (B).

Computational details (code) are available athttps://github.com/bavla/Cores/wiki/paper .

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Cores in temporal networks A longitudinal approach

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