# Problems on Large Sparse Graphs

### Payam Delgosha

### EECS Department, University of California, Berkeley

### Technical Report No. UCB/EECS-2020-166

### August 14, 2020

### http://www2.eecs.berkeley.edu/Pubs/TechRpts/2020/EECS-2020-166.pdf

In this thesis, we study two category of problems involving large sparse graphs, namely the problem of compression for graphical data, and load balancing in networks. We achieve this by employing the framework of local weak convergence, or so called the objective method. This framework provides a viewpoint which enables one to make sense of a notion of stationary stochastic processes for sparse graphs.

By employing the local weak convergence framework, we introduce a notion of entropy for probability distributions on rooted graphs. This is a generalization of the notion of entropy introduced by Bordenave and Caputo to graphs which carry marks on their vertices and edges. Such marks can represent information on real-world data. This notion of entropy can be considered as a natural counterpart for the Shannon entropy rate in the world of sparse graphical data. We illustrate this by introducing a universal compression scheme for sparse marked graphs. Furthermore, we study distributed compression of graphical data. In particular, we introduce a version of the Slepian-Wolf theorem for sparse marked graphs.

In addition to studying the problem of compression, we study the problem of load balancing in networks. We do this by modeling the problem as a hypergraph where each hyperedge represents a task carrying one unit of load, and each vertex represents a server. An allocation is a way of distributing this load. we study balanced allocations, which are roughly speaking those allocations in which no demand desires to change its allocation. Employing an extension of the local weak convergence theory to hypergraphs, we study certain asymptotic behaviors of balanced allocations, such as the asymptotic empirical load distribution at a typical server, as well as the asymptotic of the maximum load.

Problems studied in this thesis should be considered as examples showing the wide-range applicability of the local weak convergence theory and the above mentioned notion of entropy. In fact, this framework provides a viewpoint of stationary stochastic processes for sparse marked graphs. The theory of time series is the engine driving an enormous range of applications in areas such as control theory, communications, information theory and signal processing. It is to be expected that a theory of stationary stochastic processes for combinatorial structures, in particular graphs, would eventually have a similarly wide-ranging impact.

**Advisors:** Venkat Anantharam

BibTeX citation:

@phdthesis{Delgosha:EECS-2020-166, Author= {Delgosha, Payam}, Title= {Problems on Large Sparse Graphs}, School= {EECS Department, University of California, Berkeley}, Year= {2020}, Month= {Aug}, Url= {http://www2.eecs.berkeley.edu/Pubs/TechRpts/2020/EECS-2020-166.html}, Number= {UCB/EECS-2020-166}, Abstract= {In this thesis, we study two category of problems involving large sparse graphs, namely the problem of compression for graphical data, and load balancing in networks. We achieve this by employing the framework of local weak convergence, or so called the objective method. This framework provides a viewpoint which enables one to make sense of a notion of stationary stochastic processes for sparse graphs. By employing the local weak convergence framework, we introduce a notion of entropy for probability distributions on rooted graphs. This is a generalization of the notion of entropy introduced by Bordenave and Caputo to graphs which carry marks on their vertices and edges. Such marks can represent information on real-world data. This notion of entropy can be considered as a natural counterpart for the Shannon entropy rate in the world of sparse graphical data. We illustrate this by introducing a universal compression scheme for sparse marked graphs. Furthermore, we study distributed compression of graphical data. In particular, we introduce a version of the Slepian-Wolf theorem for sparse marked graphs. In addition to studying the problem of compression, we study the problem of load balancing in networks. We do this by modeling the problem as a hypergraph where each hyperedge represents a task carrying one unit of load, and each vertex represents a server. An allocation is a way of distributing this load. we study balanced allocations, which are roughly speaking those allocations in which no demand desires to change its allocation. Employing an extension of the local weak convergence theory to hypergraphs, we study certain asymptotic behaviors of balanced allocations, such as the asymptotic empirical load distribution at a typical server, as well as the asymptotic of the maximum load. Problems studied in this thesis should be considered as examples showing the wide-range applicability of the local weak convergence theory and the above mentioned notion of entropy. In fact, this framework provides a viewpoint of stationary stochastic processes for sparse marked graphs. The theory of time series is the engine driving an enormous range of applications in areas such as control theory, communications, information theory and signal processing. It is to be expected that a theory of stationary stochastic processes for combinatorial structures, in particular graphs, would eventually have a similarly wide-ranging impact.}, }

EndNote citation:

%0 Thesis %A Delgosha, Payam %T Problems on Large Sparse Graphs %I EECS Department, University of California, Berkeley %D 2020 %8 August 14 %@ UCB/EECS-2020-166 %U http://www2.eecs.berkeley.edu/Pubs/TechRpts/2020/EECS-2020-166.html %F Delgosha:EECS-2020-166