The Structure of Information Pathways in a Social Communication Network

1. The Structure of Information Pathways in a SocialCommunication Network Presented By: Under the guidance of: Tingting Xu Augustin Chainterau

2. Paper Objective • Study the temporal dynamics of communication using on-line data • Give temporal notion of ‘distance’ and ‘vector – clocks’ to formulate a temporal measure which will provide structural insights • Define the network backbone to be the sub-graph consisting of edges on which information has the potential to flow the quickest

3. Why Construct New Model • Discrete communication distributed non-uniformly over time • Direct and indirect flow of information • Discussion about recent research - has studied communication of an event-driven nature • The properties of systemic communication arguably determine much about the rate at which people in the network remain up-to-date on information about each other

4. The Present Work • Systemic communication and information pathways • Propose a framework for analyzing systemic communication based on inferring structural measures from the potential for information to flow between different nodes • Out-of-date information • Indirect paths – triangle-inequality violation

5. The Present Work • Data used here have complete histories of communication events over long periods of time • Main datasets - complete set of anonymized e-mail logs among all faculty and staff at a large university over two years • Enron e-mail corpus • The complete set of user-talk communications among admins and high-volume editors on Wikipedia • Vector clocks introduced by Lamport and refined by Mattern • Network backbone

6. Vector Clocks and Latency • Communication skeleton G • The latest viewthat v has of u at time t is denoted by • Define for all v and t • , refer as the vector clock of v at time t • Information latency is denoted by t - • An algorithm to compute the vector clocks for all nodes at all time in [0, T]

7. Latencies in Social Network Data • Consider only messages with at most c (ranging from 1 and 5) recipients • Focus on q-fraction of active e-mal users (Here q = 0.20) • For a time difference τ , we define the ball of radius τ around node v at time t, denoted Bτ(v, t), to be the set of all nodes whose latency with respect to v at time t is ≤ τ days. • For fixed t, the distribution of ball-sizes over nodes can be studied using a function ft(τ ), defined as the median value of |Bτ (v, t)| over all v

8. Open Worlds vs. Closed Worlds • Boundary specification problem – value of q-fraction [0, 1]

9. Quantifying the Strength of Weak Ties • The range of an edge , defined to be the unweighted shortest-path distance in the social network between and if were deleted • Edges of range greater than two are generally weak ties • Vector-clock analysis can provide evidence for the phenomenon that weak ties are the sources of important information to their endpoints • Define advance in ’s clock to be the sum of coordinatewise differences between before the update from and after the update from

10. Backbone Structures • Instantaneous Backbones • Define the backbone Ht at time t to be the graph on whose edge set is the collection of edges from G that are essential at time t. • An edge is essential if ’s most up-to-date view of is the result of direct communication from • Here the backbones Ht at fixed times t as instantaneous backbones, by contrast with the aggregate backbone which is based on an aggregate construction that takes all times into account.

11. Backbone Structures • An aggregate Backbone • For each edge in the communication skeleton G such that has sent ρv, w > 0 messages to over the full time interval [0, T], define the delay δv, w of the edge to be T/ ρv, w • The weighted graph Gδ obtained from the communication skeleton G by assigning a weight of δv, w to each edge • An edge in Gδ is essential if it forms the minimum-delay path between its two endpoints • Define the aggregate backbone H* to be the sub-graph of Gδ consisting only of essential edges

12. Backbone Structures • How to construct the aggregate Backbone H* • Compute a weighted shortest-paths tree rooted at each node of Gδ , using the delays as weights • The union of the edges in all these trees will be H*, by the following proposition • PROPOSITION An edge belongs to H* if and only if it lies on the minimum-delay path between some pair of nodes and • PROOF

13. Backbone Structures • How to construct the aggregate Backbone H* • Compute a weighted shortest-paths tree rooted at each node of Gδ , using the delays as weights • The union of the edges in all these trees will be H*, by the following proposition • PROPOSITION An edge belongs to H* if and only if it lies on the minimum-delay path between some pair of nodes and

14. Backbone Structures • Density and node degrees of the backbone • The backbone Ht and the aggregate backbone H* are surprisingly sparse related to a fairly dense communication skeleton G • This in other words, from the point of view of potential information flow, a significant majority of all edges in the social network are bypassed by faster indirected paths

15. Backbone Structures • Density and node degrees of the backbone • Considering the backbone also sheds further light on the role of high-degree nodes in the social network • High-degree nodes in the full communication skeleton G indeed have many incident edges in the aggregate backbone • However, the fraction of a node’s edges that are declared essential strictly decreases with degree.

16. Backbone Structures • Structure of the backbone • The backbone is trying to balance two competing objectives • Representing long range edges (recall definition of ‘range’) • Representing edges have high embeddedness and transmit information at short ranges over quick time scales • Define embeddedness of an edge to be the fraction of its endpoints’ neighbors that are common to both For an edge , let and denote the sets of neighbors of the endpoints and respectively. Define the embeddedness of to be / || • The backbone balances between two qualitatively different kinds of information flow

17. Varying Speed of Communication • Study what happens to information latencies (i.e. t - ) when each node varies the relative rates of its communication • Given a directed graph G, with a total rate for each node • Given a target set of nodes in G • Each node chooses a rate at which to communicate to each of its neighbors , subject to the constraint that • Define delays , where T is value of the time interval • Question here is that: for a given bound , can we choose rates for each node so that the median shortest-path delay between pairs in in the aggregate backbone is at most

18. Varying Speed of Communication • THEOREM The delay minimization problem defined above is NP – complete • Sketch of the proof of this theorem is in the paper • Consider simple local rules by which individuals in a network might vary rates of communication so as to influence the potential for information flow

19. Load-leveling vs. Load-concentrating • For accelerating potential information flow • Talk even more actively to one’s most frequent contacts • Load-concentrating with > 1 • or balance things out by increasing communication with the less frequent contacts? • Load-leveling with < 1 • Rescaling exponent , changing the communication rate to and then normalizing all rates from to keep its total outgoing message volume the same

20. Load-leveling vs. Load-concentrating • Extend the notion of delay to node-dependent delays which will have also a fixed delay of at each node • Total delay on a path becomes the sum of edges and node delays • As increases, there is a larger penalty for more-hop paths • The value of at which network latency is optimized decreases with * = 1 at days • The backbone becomes denser and the importance of quick indirect paths diminishes

21. Conclusions (I) • Make integral use of information about how nodes communicate over time • Develop structural measures based on the potential for information to flow • The sparse sub-graph of edges most essential to keeping people up-to-date – the backbone of the network – provides important structural insights that relate to embeddedness, the role of high-degree(i.e. hubs), and the strength of weak ties • Studied the effects on information flow as nodes vary the rate at which they communicate with others in the network using different strategies

22. Conclusions (II) • Discussions in other two datasets • The situations in sparsity of the aggregate and instantaneous backbones and the variation in node degrees are similar • Difference - the ‘core’ of active communicators is much smaller in both the Enron corpus and in Wikipedia, this makes the range of an edge in the unweighted communication skeleton harder to interpret and to correlate with other measures • Further investigation • the principles that govern the dynamics of different types of information • how these principles interact with the directed, weighted nature of social communication networks

23. Thank You 