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Marko Grobelnik, Dunja Mladenic JSI Parts of the presentation taken from the tutorial Structure and function of real-world graphs and networks by Jure Leskovec, CMU/JSI

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What are networks? few examples Network properties Small worlds Power law Long tail Network Resilience Structure of networks Applications Mining e-mail server logs Mining MSN Messenger data

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Statistics Computer systems Theory and algorithms (complex) networks Machine learning / Data mining 3

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Statistics Computer systems Theory and algorithms (complex) networks Machine learning / Data mining Social Sciences Biology Physics (complex) networks Industry & Applications Computer Science 4

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Vertex / Node

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Edge/ Link

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Vertex / Node Edge/ Link Direction

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Vertex / Node Edge/ Link Direction0.3 0.6 0.1 Probabilities

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Vertex / Node Edge/ Link Direction0.3 0.6 0.1 Probabilities in dynamic networks all the elements of the graph are changing dealing with dynamic networks is active research topic

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Query Active topic during limited time period Example of Dynamic Graph (1/3)

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On 1996-08-30 Clinton and Chicago are connected Example of Dynamic Graph (2/3)

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On 1996-10-02 Clinton and Chicago are NOT connected Example of Dynamic Graph (3/3)

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Information networks: World Wide Web: hyperlinks Citation networks Blog networks Social networks: people + interactios Organizational networks Communication networks Collaboration networks Sexual networks Collaboration networks Technological networks: Power grid Airline, road, river networks Telephone networks Internet Autonomous systems Florence families Karate club network Collaboration network Friendship network

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Biological networks metabolic networks food web neural networks gene regulatory networks Language networks Semantic networks Software networks Yeast protein interactions Semantic network Language network XFree86 network

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Directed/undirected Multi graphs (multiple edges between nodes) Hyper graphs (edges connecting multiple nodes) Bipartite graphs (e.g., papers to authors) Weighted networks Different type nodes and edges Evolving networks: Nodes and edges only added Nodes, edges added and removed

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Sociologists were first to study networks: Study of patterns of connections between people to understand functioning of the society People are nodes, interactions are edges Questionares are used to collect link data (hard to obtain, inaccurate, subjective) Typical questions: Centrality and connectivity Limited to small graphs (~10 nodes) and properties of individual nodes and edges

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Large networks (e.g., web, internet, on-line social networks) with millions of nodes Many traditional questions not useful anymore: Traditional: What happens if a node U is removed? Now: What percentage of nodes needs to be removed to affect network connectivity? Focus moves from a single node to study of statistical properties of the network as a whole Can not draw (plot) the network and examine it

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How the network looks like even if I cant look at it? Need for statistical methods and tools to quantify large networks 3 parts/goals: Statistical properties of large networks Models that help understand these properties Predict behavior of networked systems based on measured structural properties and local rules governing individual nodes

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Features common to networks of different types: Properties of static networks: Small-world effect Transitivity or clustering Degree distributions (scale free networks) Network resilience Community structure Subgraphs or motifs Temporal properties: Densification Shrinking diameter

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Six degrees of separation (Milgram 60s) Random people in Nebraska were asked to send letters to stockbrokes in Boston Letters can only be passed to first-name acquantices Only 25% letters reached the goal But they reached it in about 6 steps Measuring path lengths: Diameter (longest shortest path): max d ij Effective diameter: distance at which 90% of all connected pairs of nodes can be reached Mean geodesic (shortest) distance l

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Empirical observation for the Web-Graph is that the diameter of the Web-Graph is small relative to the size of the network this property is called Small World formally, small-world networks have diameter exponentially smaller then the size By simulation it was shown that for the Web- size of 1B pages the diameter is approx. 19 steps empirical studies confirmed the findings

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The network represents collaboration between institutions on FP5-IST projects funded by European Union there are 7886 organizations collaborating on 2786 projects in the network, each node is an organization, two organizations are connected if they collaborate on at least one project Small world properties of the collaboration network: Main connected part of the network contains 94% of the nodes Max distance between any two organizations is 7 steps meaning that any organization can be reached in up to 7 steps from any other organization Average distance between any two organizations is 3.15 steps (with standard deviation 0.38) 38% (2770) of organizations have avg. distance 3 or less

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1856 collaborations avg. distance is 1.95 max. distance is 4

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179 collaborations avg. distance is 2.42 max. distance is 4

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8 collaborations max. distance is 7

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Distribution of shortest path lengths Microsoft Messenger network 180 million people 1.3 billion edges Edge if two people exchanged at least one message in one month period Distance (Hops) Number of nodes Pick a random node, count how many nodes are at distance 1,2,3... hops 7

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Power law describes relations between the objects in the network it is very characteristic for the networks generated within some kind of social process it describes scale invariance found in many natural phenomena (including physics, biology, sociology, economy and linguistics)

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In the context of Web the power-law appears in many cases: Web pages sizes Web page connectivity Web connected components size Web page access statistics Web Browsing behavior Formally, power law describing web page degrees are: (This property has been preserved as the Web has grown)

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Degree distribution number of people a person talks to on a Microsoft Messenger Node degree Count X Highest degree

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This is not directly related to graphs, but it nicely explains the long tail effect. It shows that there is big market for niche products.

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We observe how the connectivity (length of the paths) of the network changes as the vertices get removed It is important for epidemiology Removal of vertices corresponds to vaccination Real-world networks are resilient to random attacks One has to remove all web- pages of degree > 5 to disconnect the web but this is a very small percentage of web pages Random network has better resilience to targeted attacks

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What are the building blocks (motifs) of networks? Do motifs have specific roles in networks? Network motifs detection process: Count how many times each subgraph appears Compute statistical significance for each subgraph probability of appearing in random as much as in real network 3 node motifs

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Biological networks Feed-forward loop Bi-fan motif Web graph: Feedback with two mutual diads Mutual diad Fully connected triad

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Intuition says that distances between the nodes slowly grow as the network grows (like log n ) But as the network grows the distances between nodes slowly decrease Internet Citations

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In November 1999 large scale study using AltaVista crawls in the size of over 200M nodes and 1.5B links reported bow tie structure of web links we suspect, because of the scale free nature of the Web, this structure is still preserved

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SCC - Strongly Connected component where pages can reach each other via directed paths IN consisting from pages that can reach core via directed path, but cannot be reached from the core OUT consisting from pages that can be reached from the core via directed path, but cannot reach core in a similar way TENDRILS disconnected components reachable only via directed path from IN and OUT but not from and to core

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We address the problem how to construct a taxonomy from a social network data. we adapt the approach used when dealing with text As an example we use e-mail graph in a mid size research institution ...communication records of JSI 770 people The experiments and evaluation show our approach to be useful and applicable in real life situations the approach could be easily reused in case studies (and elsewhere)

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The main contribution of the deliverable is architecture & software consisting from 5 major steps: 1.Starting with log files from the institutional e-mail server where the data include information about e-mail transactions with three fields: time, sender and the list of receivers. 2.After cleaning we get the data in the form of e-mail transactions which include e-mail addresses of sender and receiver. 3.From a set of e-mail transactions we construct a graph where vertices are e-mail addresses connected if there is a transaction between them 4.E-mail graph is transformed into a sparse matrix allowing to perform data manipulation and analysis operations 5.Sparse matrix representation of the graph is analyzed with ontology learning tools producing an ontological structure corresponding to the organizational structure of the institution where e-mails came from.

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The data is the collection of log files with e- mail transactions from local e-mail spam filter software Amavis (http://www.amavis.org/):http://www.amavis.org/ Each line of the log files denotes one event at the spam filter software We were interested in the events on successful e- mail transactions ...having information on time, sender, and list of receivers An example of successful e-mail transaction is the following line: 2005 Mar 28 13:59:05 patsy amavis[33972]: (33972-01-3) Passed CLEAN, [217.32.164.151] [193.113.30.29] ->, Message-ID:, Hits: -1.668, 6389 ms

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The log files include e-mails data from Sep 5th 2003 to Mar 28th 2005: this sums up to 12.8Gb of data. After filtering out successful e-mail transactions it remains 564Mb which contains approx. 2.7 million of successful e-mail transitions used for further processing The whole dataset contains references to approx. 45000 e-mail addresses after the data cleaning phase the number is reduced to approx. 17000 e-mail addresses out of which 770 e-mail addresses are internal from the home institution (with ijs.si domain name)

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Organizational structure of JSI produced from cleaned e-mail transactions with OntoGen in